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How do you simplify \[{(x - y)^3}?\]

Last updated date: 03rd Mar 2024
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IVSAT 2024
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Hint: To simplify the given expression we will multiply the term \[(x - y)\] three times in an algebraic mathematical process. On doing calculations we get the required answer.

Formula used: Here we will use the following indices formula:
\[{a^{m + n}} = {a^m}{a^n}\]

Complete Step by Step Solution:
We have to simplify \[{(x - y)^3}\] in a formation of addition or subtraction with variables.
Now, the term \[(x - y)\] is up to the power of \[3\].
So, we can rewrite the expression as: \[{(x - y)^{1 + 1 + 1}}\].
Now, we can use the above indices formula.
As the expression given is \[{(x - y)^3}\], it is clearly visible that we can multiply \[(x - y)\] three times to achieve the final expression.
So, we can rewrite the expression as following:
\[ \Rightarrow {(x - y)^3} = (x - y) \times (x - y) \times (x - y).\]
So, we will multiply first two \[(x - y)\]’s then the result of it will multiply by another \[(x - y)\] to get the final answer.
Therefore, the multiplication of \[(x - y) \times (x - y)\] is as following:
\[ \Rightarrow (x - y) \times (x - y) = (x - y) \times x - (x - y) \times y\].
So, after the multiplication of these terms, we get the following expression:
\[ \Rightarrow ({x^2} - xy) - (xy - {y^2})\].
Now, if we take the brackets off from the above expression, we get the following expression:
\[ \Rightarrow {x^2} - xy - xy + {y^2}\].
Now, if we arrange the middle terms and other terms accordingly, we get the following expression:
\[ \Rightarrow {x^2} - 2xy + {y^2}\].
So, we can state that :
\[ \Rightarrow {(x - y)^2} = (x - y) \times (x - y)\]
Therefore, \[{(x - y)^2} = {x^2} - 2xy + {y^2}\].
Now, to get the final value of the given expression\[{(x - y)^3}\], we have to multiply \[{(x - y)^2}\] by \[(x - y)\]
Therefore, we can write the following expression:
\[ \Rightarrow {(x - y)^3} = {(x - y)^2} \times (x - y)\].
So, we can rewrite it as following:
\[ \Rightarrow {(x - y)^3} = (x - y) \times ({x^2} - 2xy + {y^2})\].
So, if we multiply the above terms and split up, we get:
\[ \Rightarrow ({x^2} - 2xy + {y^2}) \times x - ({x^2} - 2xy + {y^2}) \times y\].
Now, if we multiply it further, we get:
\[ \Rightarrow ({x^2} \times x - 2xy \times x + {y^2} \times x) - ({x^2} \times y - 2xy \times y + {y^2} \times y)\].
Now, by doing the further simplification, we get:
\[ \Rightarrow ({x^3} - 2{x^2}y + x{y^2}) - ({x^2}y - 2x{y^2} + {y^3})\].
Now, if we take the brackets off, we get:
\[ \Rightarrow {x^3} - 2{x^2}y + x{y^2} - {x^2}y + 2x{y^2} - {y^3}\].
Now, add or subtract the terms having same degrees, we get:
\[ \Rightarrow {x^3} - 3{x^2}y + 3x{y^2} - {y^3}\].
We can rearrange the above statement in following way:
\[ \Rightarrow {x^3} - {y^3} - 3xy(x - y)\].

\[\therefore \]The answer of the question is \[{x^3} - {y^3} - 3xy(x - y)\].

Note: Points to remember:
If the power of any expression or term is \[n\], where \[n\] is any natural number, then the expression shall multiply \[n\] numbers of times.
Always expand each term in the bracket by all the other terms in the other brackets, but never multiply two or more terms in the same bracket.
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